Separatrices for non solvable dynamics on ,0
Annales de l'Institut Fourier, Volume 44 (1994) no. 2, p. 569-599

We define the separatrices for pseudogroups of diffeomorphisms of open neighbourhoods of the origin in the complex plane and prove their existence for non solvable pseudogroups (Theorem 1). This extends a result by Shcherbakov (in [21]) accurately. Our method also applies to prove the topological rigidity theorem for generic pseudogroups attributed to Shcherbakov (dans [20]).

Nous définissons les séparatrices pour les pseudo-groupes de difféomorphismes de voisinages ouverts de l’origine du plan complexe , et nous démontrons leur existence pour les pseudo-groupes non résolubles (Théorème 1). Ceci précise un résultat de Shcherbakov (dans [21]). Notre méthode permet aussi de démontrer le théorème de rigidité topologique pour les pseudo-groupes génériques attribué à Shcherbakov (dans [20]).

@article{AIF_1994__44_2_569_0,
     author = {Nakai, Isao},
     title = {Separatrices for non solvable dynamics on ${\mathbb {C}},0$},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {44},
     number = {2},
     year = {1994},
     pages = {569-599},
     doi = {10.5802/aif.1410},
     zbl = {0804.57022},
     mrnumber = {95j:58124},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1994__44_2_569_0}
}
Nakai, Isao. Separatrices for non solvable dynamics on ${\mathbb {C}},0$. Annales de l'Institut Fourier, Volume 44 (1994) no. 2, pp. 569-599. doi : 10.5802/aif.1410. http://www.numdam.org/item/AIF_1994__44_2_569_0/

[1]I.N. Baker, Fractional iteration near a fixpoint of multiplier 1, J. Australian Math. Soc., 4 (1964), 143-148. | MR 29 #2369 | Zbl 0134.05402

[2]A.F. Beardon, Iteration of Rational Functions, Graduate Texts in Math. 132, Springer-Verlag, 1991. | MR 92j:30026 | Zbl 0742.30002

[3]A.D. Brjuno, Analytic form of differential equations, Transaction Moscow Math. Soc., 25 (1971), 131-288. | MR 51 #13365 | Zbl 0272.34018

[4]C. Camacho, On the local structure of conformal mappings and holomorphic vector fields, Astérisque, 59-60 (1978), 83-84. | MR 81d:58016 | Zbl 0415.30015

[5]D. Cerveau, R. Moussu, Groupes d'automorphismes de ℂ,0 et équations différentielles y dy +...= 0, Bull. Soc. Math. France, 116 (1988). | Numdam | MR 90m:58192 | Zbl 0696.58011

[6]D. Cerveau, P. Sad, Problèmes de modules pour les formes différentielles singulières dans le plan complexe, Comment. Math. Helvetici, 61 (1986), 222-253. | MR 88f:58124 | Zbl 0604.58004

[7]J. Écalle, Les fonctions Résurgentes I-III, preprints in Université de Paris, Orsay, 1985.

[8]P. Fatou, Sur les équations fonctionnelles, Bull. S.M.F., (1919), 161-271 48 (1920), 33-94, 208-304. | JFM 47.0921.02 | Numdam | Numdam

[9]Xavier Gomez-Mont, The transverse dynamics of a holomorphic flow, Ann. Math., (1988), 49-92, 127. | MR 89d:32049 | Zbl 0639.32013

[10]Yu. S. Il'Yashenko, The topology of phase portraits of analytic differential equations in the complex projective plane, Trudy Sem Petrovsky, 4 (1978), 83-136. | MR 524528 | Zbl 0418.34007

[11]Yu.S. Il'Yashenko, The finiteness problem for limit cycles of polynomial vector fields on the plane, germs of saddle resonant vector fields and non-Hausdorff Riemann surfaces, Lecture Notes in Math. No 1060, 290-305. | MR 770249 | Zbl 0588.34024

[12]Yu.S. Il'Yashenko, Finiteness Theorems for Limit Cycles, Translations of Mathematical Monographs, AMS, 94, 1991. | MR 1133882 | MR 92k:58221 | Zbl 0743.34036

[13]T. Kimura, On the iteration of analytic functions, Funk. Equacioj, 14-3 (1971), 197-238. | MR 302876 | MR 46 #2019 | Zbl 0237.30008

[14]V.P. Kostov, Versal deformation of differential forms of degree α on a line, Funct. Anal. App., 18 (4) (1985), 335-337. | MR 775939 | MR 86g:32039 | Zbl 0573.58002

[15]J.F. Mattei, R. Moussu, Holonomie et intégrales première, Ann. Sc. Ec. Norm. Sup., 13 (1980), 469-523. | Numdam | MR 608290 | MR 83b:58005 | Zbl 0458.32005

[16]J. Martinet, Remarques sur la bifurcation Noeud-col dans le domaine complexe, Asterisque, 150-151 (1987), 131-149. | MR 923597 | MR 89d:58101 | Zbl 0655.58025

[17]I. Nakai, On toplogical types of polynomial mappings, Topology, 23, No. 1 (1984), 45-66. | MR 721451 | MR 85g:58076 | Zbl 0531.58004

[18]I. Nakai, Topology of complex webs of codimension one and geometry of projective space curves, Topology, 26 (4) (1987), 475-504. | MR 919731 | MR 89b:14010 | Zbl 0647.57018

[19]A. Lins Neto, Construction of singular holomorphic vector fields and foliations in dimension two, J. Differential Geometry, 26 (1987), 1-31. | MR 892029 | MR 88f:32047 | Zbl 0625.57012

[20]A.A Scherbakov, Topological and analytic conjugation of non commutative groups of conformal mappings, Trudy Sem. Petrovsk, 10 (1984), 170-192, 238-239. | MR 778885 | Zbl 0568.30010

[21]A.A. Scherbakov, On the density of an orbit of a pseudogroup of conformal mappings and a generalization of the Hudai-Verenov theorem, Vestnik Movskovskogo Universiteta. Mathematika, 31, No.4 (1982), 10-15. | MR 671879 | Zbl 0517.30009

[22]S.M. Voronin, Analytic classification of germs of maps (ℂ,0) →(ℂ,0) with identical linear part, Funct. Anal., 15, No.1 (1981), 1-17. | MR 609790 | MR 82h:58008 | Zbl 0463.30010

[23]S.M. Voronin, Analytic classification of pairs of involutions and its applications, Funct. Anal., 16, No.2 (1982), 94-100. | MR 659162 | MR 83j:58013 | Zbl 0521.30010

[24]M.O. Hudai-Verenov, A property of the solution of a differential equation, Mat. Sb., 56(98) : 3 (1962), 301-308. | MR 147699 | Zbl 0111.27902